Physics · Particle & Nuclear Physics · Relativistic Physics
Time Dilation Calculator
Calculates relativistic time dilation — the difference in elapsed time experienced by a moving observer versus a stationary observer — using Einstein's special relativity formula.
Calculator
Formula
t is the dilated time (coordinate time) measured by the stationary observer; t₀ is the proper time measured by the moving observer; v is the velocity of the moving observer; c is the speed of light (2.998 × 10⁸ m/s); γ (the Lorentz factor) = 1/√(1 − v²/c²) quantifies how much time stretches relative to the rest frame.
Source: Einstein, A. (1905). 'Zur Elektrodynamik bewegter Körper.' Annalen der Physik, 17, 891–921. Also: Griffiths, D.J. Introduction to Electrodynamics, 4th ed., Cambridge University Press.
How it works
Time dilation arises from the postulates of special relativity: the laws of physics are the same in all inertial reference frames, and the speed of light in a vacuum is constant for all observers. When an object moves at a significant fraction of the speed of light relative to a stationary observer, the moving object's clock appears to tick more slowly from the stationary frame. This is not an illusion — it is a real, measurable difference in elapsed time that has been confirmed by atomic clocks aboard aircraft and satellites, as well as by observations of muon decay in cosmic ray experiments.
The governing formula is t = t₀ / √(1 − v²/c²), where t₀ is the proper time — the time elapsed on the moving observer's own clock — and t is the coordinate time measured by the stationary observer. The denominator √(1 − v²/c²) is the reciprocal of the Lorentz factor γ. When v is small compared to c, γ ≈ 1 and t ≈ t₀, meaning no perceptible dilation. As v approaches c, γ grows without bound, and t can become arbitrarily large. At exactly c, the formula breaks down (division by zero), which is why massive objects cannot reach the speed of light.
Practical applications include the GPS satellite network, where atomic clocks on satellites moving at ~3.87 km/s relative to Earth's surface run approximately 7 microseconds slow per day due to special relativistic time dilation (partially offset by general relativistic effects). Particle accelerators rely on time dilation: muons created by cosmic rays in the upper atmosphere have a rest-frame half-life of ~2.2 microseconds, yet they routinely reach Earth's surface because their high velocity dilates their internal time, extending their apparent lifetime in the lab frame by a factor of ten or more.
Worked example
Problem: An astronaut travels at v = 2.4 × 10⁸ m/s (approximately 80% of the speed of light). According to the astronaut's own clock, the journey takes t₀ = 3,600 seconds (one hour). How much time has elapsed on Earth?
Step 1 — Compute v²/c²: c = 2.998 × 10⁸ m/s, so v/c = 2.4 × 10⁸ / 2.998 × 10⁸ ≈ 0.8005. Therefore v²/c² ≈ 0.6408.
Step 2 — Compute the Lorentz factor γ: γ = 1 / √(1 − 0.6408) = 1 / √(0.3592) = 1 / 0.5993 ≈ 1.6686.
Step 3 — Compute the dilated time: t = γ × t₀ = 1.6686 × 3,600 ≈ 6,007 seconds (about 1 hour and 40 minutes).
Step 4 — Compute the time difference: Δt = t − t₀ = 6,007 − 3,600 = 2,407 seconds (about 40 extra minutes pass on Earth compared to the astronaut's clock).
This is the essence of the famous twin paradox: the travelling twin returns younger than the Earth-bound twin, having experienced less time during the journey.
Limitations & notes
This calculator applies special relativistic time dilation only — it does not account for gravitational (general relativistic) time dilation, which requires a separate treatment using the Schwarzschild metric or the equivalence principle. For accurate GPS corrections or scenarios near massive objects like black holes or neutron stars, both effects must be combined. The formula assumes constant velocity in a straight line; accelerating frames introduce additional complexity. Velocities entered must be strictly less than c (2.998 × 10⁸ m/s) — the calculator will return undefined results at or above this threshold. At non-relativistic speeds (v ≪ c), the computed dilation is real but practically immeasurable without ultra-precise atomic clocks. The formula also assumes flat spacetime and does not apply in curved spacetime without extension to general relativity.
Frequently asked questions
What is the Lorentz factor and why does it matter?
The Lorentz factor γ = 1/√(1 − v²/c²) is the multiplier that quantifies all relativistic effects including time dilation, length contraction, and relativistic mass increase. A γ of 1 means no relativistic effect; at v = 0.99c, γ ≈ 7.09, meaning time passes over seven times more slowly for the moving observer than for a stationary one.
Has time dilation actually been measured experimentally?
Yes. The Hafele–Keating experiment (1971) flew atomic clocks around the world on aircraft and compared them to ground-based clocks, confirming both special and general relativistic time dilation. The decay of muons produced by cosmic rays also directly confirms time dilation — muons with a lab-frame lifetime far exceeding their rest-frame half-life reach Earth's surface precisely because of this effect.
What is the difference between proper time and coordinate time?
Proper time (t₀) is the time measured by a clock that travels with the moving object — it is the most direct, 'physical' time for that observer. Coordinate time (t) is the time measured by a stationary observer's clock in their rest frame. Special relativity shows these two quantities differ whenever relative velocity is significant, with the moving observer's proper time always being less than or equal to the coordinate time.
Why can't objects with mass travel at the speed of light?
As velocity increases toward c, the Lorentz factor γ approaches infinity. This means the energy required to accelerate a massive object also approaches infinity, making reaching c physically impossible. Additionally, at v = c the time dilation formula yields division by zero — the stationary observer would see the moving clock frozen in time entirely.
Does time dilation affect GPS satellites?
Yes. GPS satellites orbit at ~3.87 km/s, causing special relativistic time dilation that makes their clocks run about 7 microseconds slow per day relative to Earth. However, general relativistic effects due to weaker gravity at altitude cause their clocks to run about 45 microseconds fast per day. The net correction is approximately +38 microseconds per day, and GPS systems apply this correction continuously to maintain positional accuracy within meters.
Last updated: 2025-01-15 · Formula verified against primary sources.